Multiplicity (Odd Crosses vs. Even Bounces)
SAT glossary · Math
Definition
The multiplicity of a root determines the graph's behavior at that x-intercept. Odd multiplicity (1, 3, ...) means the curve CROSSES the x-axis; even multiplicity (2, 4, ...) means it merely TOUCHES and BOUNCES back without crossing. A factor raised to an even power therefore produces a zero with no sign change - the trap is assuming every zero is a crossing.
Multiplicity (Odd Crosses vs. Even Bounces) in plain English
This is really the same rule as regular 'multiplicity,' just phrased with a memorable trick: a factor raised to an ODD power (like 1, 3) means the graph CROSSES the x-axis at that zero, while a factor raised to an EVEN power (like 2, 4) means the graph merely BOUNCES off the x-axis without crossing through. Think of it like a ball hitting the ground: an odd-power zero is the ball punching straight through the floor, while an even-power zero is the ball bouncing back up. The trap to avoid: don't assume every single zero on a graph automatically means the curve crosses through — some zeros are just touch-and-go bounces, with no actual sign change happening there.
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